Write the function in the standard form a·sin(b(θ + c)) + d or a·cos(b(θ + c)) + d. Then read off: - Amplitude = |a| (vertical dilation; distance from midline to max). - Midline (vertical shift) = y = d. - Period = 2π / |b| (horizontal dilation). - Phase shift = −c (horizontal translation: right if −c > 0, left if −c < 0). If you see a·sin(bθ + φ) instead, set φ = b·c so c = φ/b, and phase shift = −φ/b. Always keep b’s sign when computing period (use |b|) and use absolute value for amplitude. Quick example: f(θ)=3 sin(4(θ − π/6)) + 2 → amplitude 3, midline y=2, period = 2π/4 = π/2, phase shift = +π/6 (right). On the AP exam keep angle mode and units consistent (CED Topic 3.6.A). For a short study review see the topic guide (https://library.fiveable.me/ap-pre-calculus/unit-3/sinusoidal-function-transformations/study-guide/1xRAbpsfqkTOU10kPQ4p) and try practice problems (https://library.fiveable.me/practice/ap-pre-calculus).

Use the standard transformed forms: f(θ) = a·sin(b(θ + c)) + d or g(θ) = a·cos(b(θ + c)) + d. What each parameter does (CED terms): - Amplitude = |a| (vertical dilation; how tall the waves are). - Midline / vertical shift = d (vertical translation of the midline). - Period = 2π / |b| (horizontal dilation; how long one cycle is). - Phase shift = −c (horizontal translation; shift right if c < 0, left if c > 0). - Note: a ≠ 0. Sine and cosine have the same transformations because cos is a sine phase-shifted by −π/2 (CED 3.6.A.1, 3.6.A.6). For AP exam work: write the model in one of those forms, identify |a|, d, 2π/|b|, and −c, and show units/labels if it’s a real context (calculator in radian mode when needed). For a focused review, see the Topic 3.6 study guide (https://library.fiveable.me/ap-pre-calculus/unit-3/sinusoidal-function-transformations/study-guide/1xRAbpsfqkTOU10kPQ4p) and practice problems (https://library.fiveable.me/practice/ap-pre-calculus).

Short answer: it’s mostly a choice of convenience—sine and cosine produce the same family of graphs because cos θ = sin(θ + π/2). Use whichever one makes identifying amplitude, midline, period, and phase shift easiest. How to pick: - If the graph at θ = 0 is at a maximum (midline + amplitude), a cosine form y = a cos(b(θ + c)) + d is usually simplest. - If the graph at θ = 0 is at the midline and rising, use sine: y = a sin(b(θ + c)) + d. - If the graph at θ = 0 is at the midline and falling, use sine with a negative a or shift by π. Remember CED facts: amplitude = |a|, period = 2π/|b|, phase shift = -c, vertical shift = d, and additive/multiplicative transforms work the same for both (CED 3.6.A.1–3.6.A.6). For AP problems keep angles in radian mode on the calculator. For extra practice and a topic review check the Fiveable study guide (https://library.fiveable.me/ap-pre-calculus/unit-3/sinusoidal-function-transformations/study-guide/1xRAbpsfqkTOU10kPQ4p) and practice problems (https://library.fiveable.me/practice/ap-pre-calculus).

They’re basically the same idea, but “phase shift” is the AP/trig way to talk about a horizontal translation of a sinusoid—and you must account for the horizontal dilation (the b inside). Concretely: - A horizontal shift of a general function f(x) by h moves every x to x − h. - For a sinusoid written a·sin(b(θ + c)) + d (CED form), the phase shift = −c (and the period = 2π/|b|). If your sinusoid is written a·sin(bθ + φ) instead, the phase shift = −φ/b. So don’t forget to divide by b when φ isn’t factored inside b(·). On the AP exam you’ll be asked to identify amplitude |a|, midline d, period 2π/|b|, and phase shift (apply the sign convention and the 1/b factor when needed). For more examples and practice, check the Topic 3.6 study guide (https://library.fiveable.me/ap-pre-calculus/unit-3/sinusoidal-function-transformations/study-guide/1xRAbpsfqkTOU10kPQ4p) and try problems at Fiveable practice (https://library.fiveable.me/practice/ap-pre-calculus).

Write f(x) in the form a·sin(b(x + c))—here a = 1 and b = 3. For y = a·sin(b(θ + c)) the period = 2π / |b| (CED 3.6.A.6). So for f(x) = sin(3x + 2), - b = 3 → period = 2π / 3. (If you want the phase shift too: rewrite 3x + 2 = 3(x + 2/3), so the phase shift is −2/3 units to the right.) Remember AP problems use radians for trig; set your calculator to radian mode on the exam. For more practice and review on sinusoidal transformations, see the Topic 3.6 study guide (https://library.fiveable.me/ap-pre-calculus/unit-3/sinusoidal-function-transformations/study-guide/1xRAbpsfqkTOU10kPQ4p) and extra practice (https://library.fiveable.me/practice/ap-pre-calculus).

Amplitude and vertical shift are different parts of y = a sin(b(θ + c)) + d (CED 3.6.A). Amplitude = |a|—how far the graph moves up/down from its midline, so it controls the “height” of peaks and troughs. Vertical shift = d—it moves the whole sine/cosine curve up or down; d is the midline (y = d). Quick example: y = 3 sin(2(θ + π/4)) − 1. Here |a| = 3 (amplitude), d = −1 (vertical shift). Midline is y = −1, so max = −1 + 3 = 2 and min = −1 − 3 = −4; range = [−4, 2]. On the AP exam you’ll be asked to identify these from equations and graphs (CED 3.6.A.6). For more practice and quick reminders, check the Topic 3.6 study guide (https://library.fiveable.me/ap-pre-calculus/unit-3/sinusoidal-function-transformations/study-guide/1xRAbpsfqkTOU10kPQ4p) and the Unit 3 overview (https://library.fiveable.me/ap-pre-calculus/unit-3).

The b multiplies the input, so it controls the horizontal dilation (period) and the graph’s “speed.” Specifically: - Period = 2π/|b|, so larger |b| → shorter period (more cycles per unit), smaller |b| → longer period (stretched horizontally). This is the horizontal dilation by factor |1/b| (CED 3.6.A.5–6). - If b is negative, sin(b(x+c)) = −sin(|b|(x+c)), so the negative sign effectively flips the wave across its midline (equivalent to multiplying the output by −1, which can be absorbed into a). - b does NOT change amplitude (that’s |a|) or the midline (d) or the phase shift amount (c still gives phase shift −c), but it does change how quickly the sine cycles repeat (frequency/ angular frequency = b). For more practice and AP-aligned explanation, check the Topic 3.6 study guide (https://library.fiveable.me/ap-pre-calculus/unit-3/sinusoidal-function-transformations/study-guide/1xRAbpsfqkTOU10kPQ4p) and thousands of practice problems (https://library.fiveable.me/practice/ap-pre-calculus).

Use the inside of the sine/cosine to read the sign. If your function is written in the CED form f(θ) = a·sin(b(θ + c)) + d (or a·cos(b(θ + c)) + d), then the phase shift = −c (in θ-units). So: - If c > 0, phase shift is −c → the graph shifts left by c. - If c < 0, phase shift is −c → the graph shifts right by |c|. If the function is instead written as sin(bθ + φ), convert: b(θ + c) = bθ + bc, so φ = bc and c = φ/b. Then phase shift = −c = −φ/b. Example quick checks: - sin(θ + π/4) → c = π/4 → shift = −π/4 (left π/4). - sin(2(θ − π/6)) = sin(2θ − π/3) → here c = −π/6 so shift = −(−π/6) = π/6 (right π/6). On the AP exam use radian mode and be consistent with the form when you compute shift. For more practice and examples see the Topic 3.6 study guide (https://library.fiveable.me/ap-pre-calculus/unit-3/sinusoidal-function-transformations/study-guide/1xRAbpsfqkTOU10kPQ4p) and lots of practice problems (https://library.fiveable.me/practice/ap-pre-calculus).

Start with y = 4 cos(2x − π) + 3 and match it to a cos(b(x + c)) + d. 1. Amplitude: |a| = |4| = 4. 2. Vertical shift / midline: d = 3 → midline y = 3 (graph moved up 3). 3. Rewrite inside: 2x − π = 2(x − π/2) so b = 2 and c = −π/2. The phase shift = −c = π/2 → shift right by π/2. 4. Period: 2π/|b| = 2π/2 = π (so one cycle fits in length π; horizontal compression by factor 1/2). 5. Reflection: a > 0 so no reflection across midline. Order of descriptive transforms (useful for sketching): start with y = cos x → horizontally compress by factor 1/2 (period becomes π) → shift right π/2 → vertically stretch by factor 4 (amplitude 4) → shift up 3. Quick check points: max when 2x−π = 0 → x = π/2 gives y = 4(1)+3 = 7; min when 2x−π = π → x = π gives y = −4+3 = −1. If you want practice on similar CED-style problems, see the Topic 3.6 study guide (https://library.fiveable.me/ap-pre-calculus/unit-3/sinusoidal-function-transformations/study-guide/1xRAbpsfqkTOU10kPQ4p) and extra problems (https://library.fiveable.me/practice/ap-pre-calculus).

Think of b as a horizontal stretch/shrink factor, not the period itself. The basic sine/cosine functions have period 2π: sin(θ+2π)=sin θ. If you replace θ by bθ, one cycle happens when bθ increases by 2π, i.e. when b(θ + P) = bθ + 2π. Solve for P: bP = 2π, so P = 2π/b. So b speeds up (b>1 → shorter period) or slows down (0https://library.fiveable.me/ap-pre-calculus/unit-3/sinusoidal-function-transformations/study-guide/1xRAbpsfqkTOU10kPQ4p) and the Unit 3 overview (https://library.fiveable.me/ap-pre-calculus/unit-3).

Start by finding the midline, amplitude, period, and phase shift directly from the graph—those four give you a, b, c, d in a·sin(b(θ + c)) + d. Steps: 1. Midline (d): average of the top and bottom values. d = (max + min)/2. 2. Amplitude (|a|): half the vertical distance. |a| = (max − min)/2. (Sign of a gives vertical reflection; check whether graph starts going up or down.) - Example: max = 5, min = −1 → midline = 2, amplitude = 3. 3. Period (T): horizontal distance for one full cycle (peak to next peak, trough to next trough, or repeating pattern). Then b = 2π / T (AP exam uses radians). 4. Phase shift (−c): find where the base sine pattern would start—for sine, use the point where the graph crosses the midline going upward. Phase shift = horizontal shift of that point from θ = 0 (so c = −(shift)). For cosine models, use the peak location. Quick tip: label one full cycle, note key points (midline crossings, peaks, troughs), then compute a, b, c, d. Practice with graphs—see the Topic 3.6 study guide (https://library.fiveable.me/ap-pre-calculus/unit-3/sinusoidal-function-transformations/study-guide/1xRAbpsfqkTOU10kPQ4p) and try problems at (https://library.fiveable.me/practice/ap-pre-calculus). For exam work, keep your calculator in radian mode.

The midline of a sinusoidal function is the horizontal line halfway between its maximum and minimum—it’s the vertical shift of the sine or cosine. In the form f(θ) = a sin(b(θ + c)) + d or g(θ) = a cos(b(θ + c)) + d the midline is y = d (CED 3.6.A.1, 3.6.A.6). How to find it: - From the formula: read off d; midline is y = d. - From a graph: compute (max + min)/2. Example: if max = 8 and min = 2, midline = (8+2)/2 = 5 so y = 5. - From data: same idea—find highest and lowest values, average them. On the AP exam you’ll often be asked to identify midline as part of amplitude/vertical shift (see FRQ 3 modeling periodic contexts). If you want a quick refresher, check the Topic 3.6 study guide (https://library.fiveable.me/ap-pre-calculus/unit-3/sinusoidal-function-transformations/study-guide/1xRAbpsfqkTOU10kPQ4p). For more practice problems, visit (https://library.fiveable.me/practice/ap-pre-calculus).

Start by identifying the four parameters the CED stresses: amplitude, midline (vertical shift), period, and phase shift. Do this from the words/values in the problem: 1. Find midline d = (max + min)/2. 2. Amplitude a = |max − min|/2. 3. Period T = time between repeating features (high-to-high or low-to-low). Then b = 2π/T. 4. Determine phase shift: choose a sine or cosine form. Cosine starts at a maximum when phase shift = 0; sine starts at midline going up. Solve for c so the model matches the given time of a known point. General form: y(t) = a sin( b( t − shift ) ) + d or y(t) = a cos( b( t − shift ) ) + d. Quick example (tides): high tide 8 m at t = 0, low tide 2 m at t = 6 h → d = 5, a = 3, period T = 12 h so b = 2π/12 = π/6. Since t=0 is a max use cosine with no shift: h(t) = 3 cos( (π/6) t ) + 5. On the AP, FRQ modeling questions expect you to show these steps and justify choices (amplitude, midline, period, phase)—see Topic 3.6 study guide for examples (https://library.fiveable.me/ap-pre-calculus/unit-3/sinusoidal-function-transformations/study-guide/1xRAbpsfqkTOU10kPQ4p). For extra practice, try problems at (https://library.fiveable.me/practice/ap-pre-calculus).

You add +3 inside the angle. In the form a·sin(b(θ + c)) + d the graph is shifted by −c units (CED 3.6.A.3 & 3.6.A.6). So a “shift left 3” means c = 3, i.e. sin(θ + 3). By contrast sin(θ − 3) would shift the graph right 3. Quick checks: - sin(θ + 3) → left 3 - sin(θ − 3) → right 3 - If you use the alternate “(θ − h)” style, then h is the right shift (θ − h) so left 3 would be θ − (−3) = θ + 3. This is a common AP Precalculus point (identify phase shift = −c); see the Topic 3.6 study guide for more examples (https://library.fiveable.me/ap-pre-calculus/unit-3/sinusoidal-function-transformations/study-guide/1xRAbpsfqkTOU10kPQ4p). For extra practice, try problems from the unit practice set (https://library.fiveable.me/practice/ap-pre-calculus).

Short answer: no—the same vertical/horizontal dilations and translations change sine and cosine in the same way (same amplitude, period, midline), but remember cosine is just a phase-shifted sine, so you may need to adjust the phase when switching between them. Why: both f(θ)=a sin(b(θ+c))+d and g(θ)=a cos(b(θ+c))+d have amplitude |a|, period 2π/|b|, midline y=d, and phase shift −c (CED 3.6.A.1 and 3.6.A.6). Since cos θ = sin(θ + π/2), a cosine graph equals a sine graph shifted horizontally by π/2, so applying the same a, b, c, d to sin vs cos only differs by that built-in phase offset. Example: a cos(bθ) = a sin(b(θ + π/(2))) . AP tip: on the exam you’ll be asked to identify amplitude, period, phase shift, and midline from either sine or cosine forms—practice translating between sin and cos forms and be comfortable with radians (calculator parts require radian mode). For a quick refresher see the Topic 3.6 study guide (https://library.fiveable.me/ap-pre-calculus/unit-3/sinusoidal-function-transformations/study-guide/1xRAbpsfqkTOU10kPQ4p) and more unit review (https://library.fiveable.me/ap-pre-calculus/unit-3). For lots of practice problems go here (https://library.fiveable.me/practice/ap-pre-calculus).